Lisa, I do not know about other states, in fact, I'd never heard of this in Colorado. However, in Virginia, the school districts submit JUST a number at the end of the census. The number is inclusive of all children counted between the ages of 5-19 within the county--whether they are public school, private school, homeschool, or out of school.
The public schools, obviously, cooperate. The school districts also coordinate to make sure that students from out of a county (district?) are counted within their proper place. All private schools of any size cooperate; I don't recall if there are very small ones that may be more reluctant. The school district can also ask your neighbors (and do!) if they know of any children that aren't being counted. It is my understanding that someone else can advise your district about your children and they can be counted that way.
A few months ago, I had a 20 minute or so VERY friendly discussion with our local school administrator in charge of this on this exact topic. It was not within the context of us hsing though, just a general conversation that came about.
Someone else commented about the expense. My district HIRES a part time person at 25 hours a week for ~ three months to coordinate all the info and come up with the final head count.